Hello Pierre,
I enjoyed your visualization. However, I think the factor of 2 is coming from
the fact that the seperation between the plates in your description is not d2-d1
rather it is r2-r1. r2=d2/2 and r1=d1/2. If you redo the calculation using
r2-r1 as plane seperation rather then d2 -d1 you will have an extra factor of 2
in the nominator. Then you will see your calculation will be consistent with
Johnson's.

I made once the effort to demonstrate Johnson via capacitance my-myself,
here is my demonstration and conclusion

- picture the via being a cylinder diameter=D1
- picture a second cylinder wrapped around the first one D2>D1
- this second one represents power/gnd planes
- both cylinders are iso-potential
- un-fold both cylinders and you end up with two parallel plates.
- the separation between the plates is the via clearance
1270-711(D2)
- the height of both cylinders is the board thickness (T)
- the width of the cylinder is the via circumference times
pi=3.1416... (pi*D1).

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